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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.distribution;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.exception.NotStrictlyPositiveException;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.special.Gamma;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.random.RandomGenerator;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.random.Well19937c;<a name="line.24"></a>
<FONT color="green">025</FONT>    <a name="line.25"></a>
<FONT color="green">026</FONT>    /**<a name="line.26"></a>
<FONT color="green">027</FONT>     * Implementation of the Gamma distribution.<a name="line.27"></a>
<FONT color="green">028</FONT>     *<a name="line.28"></a>
<FONT color="green">029</FONT>     * @see &lt;a href="http://en.wikipedia.org/wiki/Gamma_distribution"&gt;Gamma distribution (Wikipedia)&lt;/a&gt;<a name="line.29"></a>
<FONT color="green">030</FONT>     * @see &lt;a href="http://mathworld.wolfram.com/GammaDistribution.html"&gt;Gamma distribution (MathWorld)&lt;/a&gt;<a name="line.30"></a>
<FONT color="green">031</FONT>     * @version $Id: GammaDistribution.java 1422195 2012-12-15 06:45:18Z psteitz $<a name="line.31"></a>
<FONT color="green">032</FONT>     */<a name="line.32"></a>
<FONT color="green">033</FONT>    public class GammaDistribution extends AbstractRealDistribution {<a name="line.33"></a>
<FONT color="green">034</FONT>        /**<a name="line.34"></a>
<FONT color="green">035</FONT>         * Default inverse cumulative probability accuracy.<a name="line.35"></a>
<FONT color="green">036</FONT>         * @since 2.1<a name="line.36"></a>
<FONT color="green">037</FONT>         */<a name="line.37"></a>
<FONT color="green">038</FONT>        public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;<a name="line.38"></a>
<FONT color="green">039</FONT>        /** Serializable version identifier. */<a name="line.39"></a>
<FONT color="green">040</FONT>        private static final long serialVersionUID = 20120524L;<a name="line.40"></a>
<FONT color="green">041</FONT>        /** The shape parameter. */<a name="line.41"></a>
<FONT color="green">042</FONT>        private final double shape;<a name="line.42"></a>
<FONT color="green">043</FONT>        /** The scale parameter. */<a name="line.43"></a>
<FONT color="green">044</FONT>        private final double scale;<a name="line.44"></a>
<FONT color="green">045</FONT>        /**<a name="line.45"></a>
<FONT color="green">046</FONT>         * The constant value of {@code shape + g + 0.5}, where {@code g} is the<a name="line.46"></a>
<FONT color="green">047</FONT>         * Lanczos constant {@link Gamma#LANCZOS_G}.<a name="line.47"></a>
<FONT color="green">048</FONT>         */<a name="line.48"></a>
<FONT color="green">049</FONT>        private final double shiftedShape;<a name="line.49"></a>
<FONT color="green">050</FONT>        /**<a name="line.50"></a>
<FONT color="green">051</FONT>         * The constant value of<a name="line.51"></a>
<FONT color="green">052</FONT>         * {@code shape / scale * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape)},<a name="line.52"></a>
<FONT color="green">053</FONT>         * where {@code L(shape)} is the Lanczos approximation returned by<a name="line.53"></a>
<FONT color="green">054</FONT>         * {@link Gamma#lanczos(double)}. This prefactor is used in<a name="line.54"></a>
<FONT color="green">055</FONT>         * {@link #density(double)}, when no overflow occurs with the natural<a name="line.55"></a>
<FONT color="green">056</FONT>         * calculation.<a name="line.56"></a>
<FONT color="green">057</FONT>         */<a name="line.57"></a>
<FONT color="green">058</FONT>        private final double densityPrefactor1;<a name="line.58"></a>
<FONT color="green">059</FONT>        /**<a name="line.59"></a>
<FONT color="green">060</FONT>         * The constant value of<a name="line.60"></a>
<FONT color="green">061</FONT>         * {@code shape * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape)},<a name="line.61"></a>
<FONT color="green">062</FONT>         * where {@code L(shape)} is the Lanczos approximation returned by<a name="line.62"></a>
<FONT color="green">063</FONT>         * {@link Gamma#lanczos(double)}. This prefactor is used in<a name="line.63"></a>
<FONT color="green">064</FONT>         * {@link #density(double)}, when overflow occurs with the natural<a name="line.64"></a>
<FONT color="green">065</FONT>         * calculation.<a name="line.65"></a>
<FONT color="green">066</FONT>         */<a name="line.66"></a>
<FONT color="green">067</FONT>        private final double densityPrefactor2;<a name="line.67"></a>
<FONT color="green">068</FONT>        /**<a name="line.68"></a>
<FONT color="green">069</FONT>         * Lower bound on {@code y = x / scale} for the selection of the computation<a name="line.69"></a>
<FONT color="green">070</FONT>         * method in {@link #density(double)}. For {@code y &lt;= minY}, the natural<a name="line.70"></a>
<FONT color="green">071</FONT>         * calculation overflows.<a name="line.71"></a>
<FONT color="green">072</FONT>         */<a name="line.72"></a>
<FONT color="green">073</FONT>        private final double minY;<a name="line.73"></a>
<FONT color="green">074</FONT>        /**<a name="line.74"></a>
<FONT color="green">075</FONT>         * Upper bound on {@code log(y)} ({@code y = x / scale}) for the selection<a name="line.75"></a>
<FONT color="green">076</FONT>         * of the computation method in {@link #density(double)}. For<a name="line.76"></a>
<FONT color="green">077</FONT>         * {@code log(y) &gt;= maxLogY}, the natural calculation overflows.<a name="line.77"></a>
<FONT color="green">078</FONT>         */<a name="line.78"></a>
<FONT color="green">079</FONT>        private final double maxLogY;<a name="line.79"></a>
<FONT color="green">080</FONT>        /** Inverse cumulative probability accuracy. */<a name="line.80"></a>
<FONT color="green">081</FONT>        private final double solverAbsoluteAccuracy;<a name="line.81"></a>
<FONT color="green">082</FONT>    <a name="line.82"></a>
<FONT color="green">083</FONT>        /**<a name="line.83"></a>
<FONT color="green">084</FONT>         * Creates a new gamma distribution with specified values of the shape and<a name="line.84"></a>
<FONT color="green">085</FONT>         * scale parameters.<a name="line.85"></a>
<FONT color="green">086</FONT>         *<a name="line.86"></a>
<FONT color="green">087</FONT>         * @param shape the shape parameter<a name="line.87"></a>
<FONT color="green">088</FONT>         * @param scale the scale parameter<a name="line.88"></a>
<FONT color="green">089</FONT>         * @throws NotStrictlyPositiveException if {@code shape &lt;= 0} or<a name="line.89"></a>
<FONT color="green">090</FONT>         * {@code scale &lt;= 0}.<a name="line.90"></a>
<FONT color="green">091</FONT>         */<a name="line.91"></a>
<FONT color="green">092</FONT>        public GammaDistribution(double shape, double scale) throws NotStrictlyPositiveException {<a name="line.92"></a>
<FONT color="green">093</FONT>            this(shape, scale, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);<a name="line.93"></a>
<FONT color="green">094</FONT>        }<a name="line.94"></a>
<FONT color="green">095</FONT>    <a name="line.95"></a>
<FONT color="green">096</FONT>        /**<a name="line.96"></a>
<FONT color="green">097</FONT>         * Creates a new gamma distribution with specified values of the shape and<a name="line.97"></a>
<FONT color="green">098</FONT>         * scale parameters.<a name="line.98"></a>
<FONT color="green">099</FONT>         *<a name="line.99"></a>
<FONT color="green">100</FONT>         * @param shape the shape parameter<a name="line.100"></a>
<FONT color="green">101</FONT>         * @param scale the scale parameter<a name="line.101"></a>
<FONT color="green">102</FONT>         * @param inverseCumAccuracy the maximum absolute error in inverse<a name="line.102"></a>
<FONT color="green">103</FONT>         * cumulative probability estimates (defaults to<a name="line.103"></a>
<FONT color="green">104</FONT>         * {@link #DEFAULT_INVERSE_ABSOLUTE_ACCURACY}).<a name="line.104"></a>
<FONT color="green">105</FONT>         * @throws NotStrictlyPositiveException if {@code shape &lt;= 0} or<a name="line.105"></a>
<FONT color="green">106</FONT>         * {@code scale &lt;= 0}.<a name="line.106"></a>
<FONT color="green">107</FONT>         * @since 2.1<a name="line.107"></a>
<FONT color="green">108</FONT>         */<a name="line.108"></a>
<FONT color="green">109</FONT>        public GammaDistribution(double shape, double scale, double inverseCumAccuracy)<a name="line.109"></a>
<FONT color="green">110</FONT>            throws NotStrictlyPositiveException {<a name="line.110"></a>
<FONT color="green">111</FONT>            this(new Well19937c(), shape, scale, inverseCumAccuracy);<a name="line.111"></a>
<FONT color="green">112</FONT>        }<a name="line.112"></a>
<FONT color="green">113</FONT>    <a name="line.113"></a>
<FONT color="green">114</FONT>        /**<a name="line.114"></a>
<FONT color="green">115</FONT>         * Creates a Gamma distribution.<a name="line.115"></a>
<FONT color="green">116</FONT>         *<a name="line.116"></a>
<FONT color="green">117</FONT>         * @param rng Random number generator.<a name="line.117"></a>
<FONT color="green">118</FONT>         * @param shape the shape parameter<a name="line.118"></a>
<FONT color="green">119</FONT>         * @param scale the scale parameter<a name="line.119"></a>
<FONT color="green">120</FONT>         * @param inverseCumAccuracy the maximum absolute error in inverse<a name="line.120"></a>
<FONT color="green">121</FONT>         * cumulative probability estimates (defaults to<a name="line.121"></a>
<FONT color="green">122</FONT>         * {@link #DEFAULT_INVERSE_ABSOLUTE_ACCURACY}).<a name="line.122"></a>
<FONT color="green">123</FONT>         * @throws NotStrictlyPositiveException if {@code shape &lt;= 0} or<a name="line.123"></a>
<FONT color="green">124</FONT>         * {@code scale &lt;= 0}.<a name="line.124"></a>
<FONT color="green">125</FONT>         * @since 3.1<a name="line.125"></a>
<FONT color="green">126</FONT>         */<a name="line.126"></a>
<FONT color="green">127</FONT>        public GammaDistribution(RandomGenerator rng,<a name="line.127"></a>
<FONT color="green">128</FONT>                                 double shape,<a name="line.128"></a>
<FONT color="green">129</FONT>                                 double scale,<a name="line.129"></a>
<FONT color="green">130</FONT>                                 double inverseCumAccuracy)<a name="line.130"></a>
<FONT color="green">131</FONT>            throws NotStrictlyPositiveException {<a name="line.131"></a>
<FONT color="green">132</FONT>            super(rng);<a name="line.132"></a>
<FONT color="green">133</FONT>    <a name="line.133"></a>
<FONT color="green">134</FONT>            if (shape &lt;= 0) {<a name="line.134"></a>
<FONT color="green">135</FONT>                throw new NotStrictlyPositiveException(LocalizedFormats.SHAPE, shape);<a name="line.135"></a>
<FONT color="green">136</FONT>            }<a name="line.136"></a>
<FONT color="green">137</FONT>            if (scale &lt;= 0) {<a name="line.137"></a>
<FONT color="green">138</FONT>                throw new NotStrictlyPositiveException(LocalizedFormats.SCALE, scale);<a name="line.138"></a>
<FONT color="green">139</FONT>            }<a name="line.139"></a>
<FONT color="green">140</FONT>    <a name="line.140"></a>
<FONT color="green">141</FONT>            this.shape = shape;<a name="line.141"></a>
<FONT color="green">142</FONT>            this.scale = scale;<a name="line.142"></a>
<FONT color="green">143</FONT>            this.solverAbsoluteAccuracy = inverseCumAccuracy;<a name="line.143"></a>
<FONT color="green">144</FONT>            this.shiftedShape = shape + Gamma.LANCZOS_G + 0.5;<a name="line.144"></a>
<FONT color="green">145</FONT>            final double aux = FastMath.E / (2.0 * FastMath.PI * shiftedShape);<a name="line.145"></a>
<FONT color="green">146</FONT>            this.densityPrefactor2 = shape * FastMath.sqrt(aux) / Gamma.lanczos(shape);<a name="line.146"></a>
<FONT color="green">147</FONT>            this.densityPrefactor1 = this.densityPrefactor2 / scale *<a name="line.147"></a>
<FONT color="green">148</FONT>                    FastMath.pow(shiftedShape, -shape) *<a name="line.148"></a>
<FONT color="green">149</FONT>                    FastMath.exp(shape + Gamma.LANCZOS_G);<a name="line.149"></a>
<FONT color="green">150</FONT>            this.minY = shape + Gamma.LANCZOS_G - FastMath.log(Double.MAX_VALUE);<a name="line.150"></a>
<FONT color="green">151</FONT>            this.maxLogY = FastMath.log(Double.MAX_VALUE) / (shape - 1.0);<a name="line.151"></a>
<FONT color="green">152</FONT>        }<a name="line.152"></a>
<FONT color="green">153</FONT>    <a name="line.153"></a>
<FONT color="green">154</FONT>        /**<a name="line.154"></a>
<FONT color="green">155</FONT>         * Returns the shape parameter of {@code this} distribution.<a name="line.155"></a>
<FONT color="green">156</FONT>         *<a name="line.156"></a>
<FONT color="green">157</FONT>         * @return the shape parameter<a name="line.157"></a>
<FONT color="green">158</FONT>         * @deprecated as of version 3.1, {@link #getShape()} should be preferred.<a name="line.158"></a>
<FONT color="green">159</FONT>         * This method will be removed in version 4.0.<a name="line.159"></a>
<FONT color="green">160</FONT>         */<a name="line.160"></a>
<FONT color="green">161</FONT>        @Deprecated<a name="line.161"></a>
<FONT color="green">162</FONT>        public double getAlpha() {<a name="line.162"></a>
<FONT color="green">163</FONT>            return shape;<a name="line.163"></a>
<FONT color="green">164</FONT>        }<a name="line.164"></a>
<FONT color="green">165</FONT>    <a name="line.165"></a>
<FONT color="green">166</FONT>        /**<a name="line.166"></a>
<FONT color="green">167</FONT>         * Returns the shape parameter of {@code this} distribution.<a name="line.167"></a>
<FONT color="green">168</FONT>         *<a name="line.168"></a>
<FONT color="green">169</FONT>         * @return the shape parameter<a name="line.169"></a>
<FONT color="green">170</FONT>         * @since 3.1<a name="line.170"></a>
<FONT color="green">171</FONT>         */<a name="line.171"></a>
<FONT color="green">172</FONT>        public double getShape() {<a name="line.172"></a>
<FONT color="green">173</FONT>            return shape;<a name="line.173"></a>
<FONT color="green">174</FONT>        }<a name="line.174"></a>
<FONT color="green">175</FONT>    <a name="line.175"></a>
<FONT color="green">176</FONT>        /**<a name="line.176"></a>
<FONT color="green">177</FONT>         * Returns the scale parameter of {@code this} distribution.<a name="line.177"></a>
<FONT color="green">178</FONT>         *<a name="line.178"></a>
<FONT color="green">179</FONT>         * @return the scale parameter<a name="line.179"></a>
<FONT color="green">180</FONT>         * @deprecated as of version 3.1, {@link #getScale()} should be preferred.<a name="line.180"></a>
<FONT color="green">181</FONT>         * This method will be removed in version 4.0.<a name="line.181"></a>
<FONT color="green">182</FONT>         */<a name="line.182"></a>
<FONT color="green">183</FONT>        @Deprecated<a name="line.183"></a>
<FONT color="green">184</FONT>        public double getBeta() {<a name="line.184"></a>
<FONT color="green">185</FONT>            return scale;<a name="line.185"></a>
<FONT color="green">186</FONT>        }<a name="line.186"></a>
<FONT color="green">187</FONT>    <a name="line.187"></a>
<FONT color="green">188</FONT>        /**<a name="line.188"></a>
<FONT color="green">189</FONT>         * Returns the scale parameter of {@code this} distribution.<a name="line.189"></a>
<FONT color="green">190</FONT>         *<a name="line.190"></a>
<FONT color="green">191</FONT>         * @return the scale parameter<a name="line.191"></a>
<FONT color="green">192</FONT>         * @since 3.1<a name="line.192"></a>
<FONT color="green">193</FONT>         */<a name="line.193"></a>
<FONT color="green">194</FONT>        public double getScale() {<a name="line.194"></a>
<FONT color="green">195</FONT>            return scale;<a name="line.195"></a>
<FONT color="green">196</FONT>        }<a name="line.196"></a>
<FONT color="green">197</FONT>    <a name="line.197"></a>
<FONT color="green">198</FONT>        /** {@inheritDoc} */<a name="line.198"></a>
<FONT color="green">199</FONT>        public double density(double x) {<a name="line.199"></a>
<FONT color="green">200</FONT>           /* The present method must return the value of<a name="line.200"></a>
<FONT color="green">201</FONT>            *<a name="line.201"></a>
<FONT color="green">202</FONT>            *     1       x a     - x<a name="line.202"></a>
<FONT color="green">203</FONT>            * ---------- (-)  exp(---)<a name="line.203"></a>
<FONT color="green">204</FONT>            * x Gamma(a)  b        b<a name="line.204"></a>
<FONT color="green">205</FONT>            *<a name="line.205"></a>
<FONT color="green">206</FONT>            * where a is the shape parameter, and b the scale parameter.<a name="line.206"></a>
<FONT color="green">207</FONT>            * Substituting the Lanczos approximation of Gamma(a) leads to the<a name="line.207"></a>
<FONT color="green">208</FONT>            * following expression of the density<a name="line.208"></a>
<FONT color="green">209</FONT>            *<a name="line.209"></a>
<FONT color="green">210</FONT>            * a              e            1         y      a<a name="line.210"></a>
<FONT color="green">211</FONT>            * - sqrt(------------------) ---- (-----------)  exp(a - y + g),<a name="line.211"></a>
<FONT color="green">212</FONT>            * x      2 pi (a + g + 0.5)  L(a)  a + g + 0.5<a name="line.212"></a>
<FONT color="green">213</FONT>            *<a name="line.213"></a>
<FONT color="green">214</FONT>            * where y = x / b. The above formula is the "natural" computation, which<a name="line.214"></a>
<FONT color="green">215</FONT>            * is implemented when no overflow is likely to occur. If overflow occurs<a name="line.215"></a>
<FONT color="green">216</FONT>            * with the natural computation, the following identity is used. It is<a name="line.216"></a>
<FONT color="green">217</FONT>            * based on the BOOST library<a name="line.217"></a>
<FONT color="green">218</FONT>            * http://www.boost.org/doc/libs/1_35_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma.html<a name="line.218"></a>
<FONT color="green">219</FONT>            * Formula (15) needs adaptations, which are detailed below.<a name="line.219"></a>
<FONT color="green">220</FONT>            *<a name="line.220"></a>
<FONT color="green">221</FONT>            *       y      a<a name="line.221"></a>
<FONT color="green">222</FONT>            * (-----------)  exp(a - y + g)<a name="line.222"></a>
<FONT color="green">223</FONT>            *  a + g + 0.5<a name="line.223"></a>
<FONT color="green">224</FONT>            *                              y - a - g - 0.5    y (g + 0.5)<a name="line.224"></a>
<FONT color="green">225</FONT>            *               = exp(a log1pm(---------------) - ----------- + g),<a name="line.225"></a>
<FONT color="green">226</FONT>            *                                a + g + 0.5      a + g + 0.5<a name="line.226"></a>
<FONT color="green">227</FONT>            *<a name="line.227"></a>
<FONT color="green">228</FONT>            *  where log1pm(z) = log(1 + z) - z. Therefore, the value to be<a name="line.228"></a>
<FONT color="green">229</FONT>            *  returned is<a name="line.229"></a>
<FONT color="green">230</FONT>            *<a name="line.230"></a>
<FONT color="green">231</FONT>            * a              e            1<a name="line.231"></a>
<FONT color="green">232</FONT>            * - sqrt(------------------) ----<a name="line.232"></a>
<FONT color="green">233</FONT>            * x      2 pi (a + g + 0.5)  L(a)<a name="line.233"></a>
<FONT color="green">234</FONT>            *                              y - a - g - 0.5    y (g + 0.5)<a name="line.234"></a>
<FONT color="green">235</FONT>            *               * exp(a log1pm(---------------) - ----------- + g).<a name="line.235"></a>
<FONT color="green">236</FONT>            *                                a + g + 0.5      a + g + 0.5<a name="line.236"></a>
<FONT color="green">237</FONT>            */<a name="line.237"></a>
<FONT color="green">238</FONT>            if (x &lt; 0) {<a name="line.238"></a>
<FONT color="green">239</FONT>                return 0;<a name="line.239"></a>
<FONT color="green">240</FONT>            }<a name="line.240"></a>
<FONT color="green">241</FONT>            final double y = x / scale;<a name="line.241"></a>
<FONT color="green">242</FONT>            if ((y &lt;= minY) || (FastMath.log(y) &gt;= maxLogY)) {<a name="line.242"></a>
<FONT color="green">243</FONT>                /*<a name="line.243"></a>
<FONT color="green">244</FONT>                 * Overflow.<a name="line.244"></a>
<FONT color="green">245</FONT>                 */<a name="line.245"></a>
<FONT color="green">246</FONT>                final double aux1 = (y - shiftedShape) / shiftedShape;<a name="line.246"></a>
<FONT color="green">247</FONT>                final double aux2 = shape * (FastMath.log1p(aux1) - aux1);<a name="line.247"></a>
<FONT color="green">248</FONT>                final double aux3 = -y * (Gamma.LANCZOS_G + 0.5) / shiftedShape +<a name="line.248"></a>
<FONT color="green">249</FONT>                        Gamma.LANCZOS_G + aux2;<a name="line.249"></a>
<FONT color="green">250</FONT>                return densityPrefactor2 / x * FastMath.exp(aux3);<a name="line.250"></a>
<FONT color="green">251</FONT>            }<a name="line.251"></a>
<FONT color="green">252</FONT>            /*<a name="line.252"></a>
<FONT color="green">253</FONT>             * Natural calculation.<a name="line.253"></a>
<FONT color="green">254</FONT>             */<a name="line.254"></a>
<FONT color="green">255</FONT>            return densityPrefactor1  * FastMath.exp(-y) *<a name="line.255"></a>
<FONT color="green">256</FONT>                    FastMath.pow(y, shape - 1);<a name="line.256"></a>
<FONT color="green">257</FONT>        }<a name="line.257"></a>
<FONT color="green">258</FONT>    <a name="line.258"></a>
<FONT color="green">259</FONT>        /**<a name="line.259"></a>
<FONT color="green">260</FONT>         * {@inheritDoc}<a name="line.260"></a>
<FONT color="green">261</FONT>         *<a name="line.261"></a>
<FONT color="green">262</FONT>         * The implementation of this method is based on:<a name="line.262"></a>
<FONT color="green">263</FONT>         * &lt;ul&gt;<a name="line.263"></a>
<FONT color="green">264</FONT>         *  &lt;li&gt;<a name="line.264"></a>
<FONT color="green">265</FONT>         *   &lt;a href="http://mathworld.wolfram.com/Chi-SquaredDistribution.html"&gt;<a name="line.265"></a>
<FONT color="green">266</FONT>         *    Chi-Squared Distribution&lt;/a&gt;, equation (9).<a name="line.266"></a>
<FONT color="green">267</FONT>         *  &lt;/li&gt;<a name="line.267"></a>
<FONT color="green">268</FONT>         *  &lt;li&gt;Casella, G., &amp; Berger, R. (1990). &lt;i&gt;Statistical Inference&lt;/i&gt;.<a name="line.268"></a>
<FONT color="green">269</FONT>         *    Belmont, CA: Duxbury Press.<a name="line.269"></a>
<FONT color="green">270</FONT>         *  &lt;/li&gt;<a name="line.270"></a>
<FONT color="green">271</FONT>         * &lt;/ul&gt;<a name="line.271"></a>
<FONT color="green">272</FONT>         */<a name="line.272"></a>
<FONT color="green">273</FONT>        public double cumulativeProbability(double x) {<a name="line.273"></a>
<FONT color="green">274</FONT>            double ret;<a name="line.274"></a>
<FONT color="green">275</FONT>    <a name="line.275"></a>
<FONT color="green">276</FONT>            if (x &lt;= 0) {<a name="line.276"></a>
<FONT color="green">277</FONT>                ret = 0;<a name="line.277"></a>
<FONT color="green">278</FONT>            } else {<a name="line.278"></a>
<FONT color="green">279</FONT>                ret = Gamma.regularizedGammaP(shape, x / scale);<a name="line.279"></a>
<FONT color="green">280</FONT>            }<a name="line.280"></a>
<FONT color="green">281</FONT>    <a name="line.281"></a>
<FONT color="green">282</FONT>            return ret;<a name="line.282"></a>
<FONT color="green">283</FONT>        }<a name="line.283"></a>
<FONT color="green">284</FONT>    <a name="line.284"></a>
<FONT color="green">285</FONT>        /** {@inheritDoc} */<a name="line.285"></a>
<FONT color="green">286</FONT>        @Override<a name="line.286"></a>
<FONT color="green">287</FONT>        protected double getSolverAbsoluteAccuracy() {<a name="line.287"></a>
<FONT color="green">288</FONT>            return solverAbsoluteAccuracy;<a name="line.288"></a>
<FONT color="green">289</FONT>        }<a name="line.289"></a>
<FONT color="green">290</FONT>    <a name="line.290"></a>
<FONT color="green">291</FONT>        /**<a name="line.291"></a>
<FONT color="green">292</FONT>         * {@inheritDoc}<a name="line.292"></a>
<FONT color="green">293</FONT>         *<a name="line.293"></a>
<FONT color="green">294</FONT>         * For shape parameter {@code alpha} and scale parameter {@code beta}, the<a name="line.294"></a>
<FONT color="green">295</FONT>         * mean is {@code alpha * beta}.<a name="line.295"></a>
<FONT color="green">296</FONT>         */<a name="line.296"></a>
<FONT color="green">297</FONT>        public double getNumericalMean() {<a name="line.297"></a>
<FONT color="green">298</FONT>            return shape * scale;<a name="line.298"></a>
<FONT color="green">299</FONT>        }<a name="line.299"></a>
<FONT color="green">300</FONT>    <a name="line.300"></a>
<FONT color="green">301</FONT>        /**<a name="line.301"></a>
<FONT color="green">302</FONT>         * {@inheritDoc}<a name="line.302"></a>
<FONT color="green">303</FONT>         *<a name="line.303"></a>
<FONT color="green">304</FONT>         * For shape parameter {@code alpha} and scale parameter {@code beta}, the<a name="line.304"></a>
<FONT color="green">305</FONT>         * variance is {@code alpha * beta^2}.<a name="line.305"></a>
<FONT color="green">306</FONT>         *<a name="line.306"></a>
<FONT color="green">307</FONT>         * @return {@inheritDoc}<a name="line.307"></a>
<FONT color="green">308</FONT>         */<a name="line.308"></a>
<FONT color="green">309</FONT>        public double getNumericalVariance() {<a name="line.309"></a>
<FONT color="green">310</FONT>            return shape * scale * scale;<a name="line.310"></a>
<FONT color="green">311</FONT>        }<a name="line.311"></a>
<FONT color="green">312</FONT>    <a name="line.312"></a>
<FONT color="green">313</FONT>        /**<a name="line.313"></a>
<FONT color="green">314</FONT>         * {@inheritDoc}<a name="line.314"></a>
<FONT color="green">315</FONT>         *<a name="line.315"></a>
<FONT color="green">316</FONT>         * The lower bound of the support is always 0 no matter the parameters.<a name="line.316"></a>
<FONT color="green">317</FONT>         *<a name="line.317"></a>
<FONT color="green">318</FONT>         * @return lower bound of the support (always 0)<a name="line.318"></a>
<FONT color="green">319</FONT>         */<a name="line.319"></a>
<FONT color="green">320</FONT>        public double getSupportLowerBound() {<a name="line.320"></a>
<FONT color="green">321</FONT>            return 0;<a name="line.321"></a>
<FONT color="green">322</FONT>        }<a name="line.322"></a>
<FONT color="green">323</FONT>    <a name="line.323"></a>
<FONT color="green">324</FONT>        /**<a name="line.324"></a>
<FONT color="green">325</FONT>         * {@inheritDoc}<a name="line.325"></a>
<FONT color="green">326</FONT>         *<a name="line.326"></a>
<FONT color="green">327</FONT>         * The upper bound of the support is always positive infinity<a name="line.327"></a>
<FONT color="green">328</FONT>         * no matter the parameters.<a name="line.328"></a>
<FONT color="green">329</FONT>         *<a name="line.329"></a>
<FONT color="green">330</FONT>         * @return upper bound of the support (always Double.POSITIVE_INFINITY)<a name="line.330"></a>
<FONT color="green">331</FONT>         */<a name="line.331"></a>
<FONT color="green">332</FONT>        public double getSupportUpperBound() {<a name="line.332"></a>
<FONT color="green">333</FONT>            return Double.POSITIVE_INFINITY;<a name="line.333"></a>
<FONT color="green">334</FONT>        }<a name="line.334"></a>
<FONT color="green">335</FONT>    <a name="line.335"></a>
<FONT color="green">336</FONT>        /** {@inheritDoc} */<a name="line.336"></a>
<FONT color="green">337</FONT>        public boolean isSupportLowerBoundInclusive() {<a name="line.337"></a>
<FONT color="green">338</FONT>            return true;<a name="line.338"></a>
<FONT color="green">339</FONT>        }<a name="line.339"></a>
<FONT color="green">340</FONT>    <a name="line.340"></a>
<FONT color="green">341</FONT>        /** {@inheritDoc} */<a name="line.341"></a>
<FONT color="green">342</FONT>        public boolean isSupportUpperBoundInclusive() {<a name="line.342"></a>
<FONT color="green">343</FONT>            return false;<a name="line.343"></a>
<FONT color="green">344</FONT>        }<a name="line.344"></a>
<FONT color="green">345</FONT>    <a name="line.345"></a>
<FONT color="green">346</FONT>        /**<a name="line.346"></a>
<FONT color="green">347</FONT>         * {@inheritDoc}<a name="line.347"></a>
<FONT color="green">348</FONT>         *<a name="line.348"></a>
<FONT color="green">349</FONT>         * The support of this distribution is connected.<a name="line.349"></a>
<FONT color="green">350</FONT>         *<a name="line.350"></a>
<FONT color="green">351</FONT>         * @return {@code true}<a name="line.351"></a>
<FONT color="green">352</FONT>         */<a name="line.352"></a>
<FONT color="green">353</FONT>        public boolean isSupportConnected() {<a name="line.353"></a>
<FONT color="green">354</FONT>            return true;<a name="line.354"></a>
<FONT color="green">355</FONT>        }<a name="line.355"></a>
<FONT color="green">356</FONT>    <a name="line.356"></a>
<FONT color="green">357</FONT>        /**<a name="line.357"></a>
<FONT color="green">358</FONT>         * &lt;p&gt;This implementation uses the following algorithms: &lt;/p&gt;<a name="line.358"></a>
<FONT color="green">359</FONT>         *<a name="line.359"></a>
<FONT color="green">360</FONT>         * &lt;p&gt;For 0 &lt; shape &lt; 1: &lt;br/&gt;<a name="line.360"></a>
<FONT color="green">361</FONT>         * Ahrens, J. H. and Dieter, U., &lt;i&gt;Computer methods for<a name="line.361"></a>
<FONT color="green">362</FONT>         * sampling from gamma, beta, Poisson and binomial distributions.&lt;/i&gt;<a name="line.362"></a>
<FONT color="green">363</FONT>         * Computing, 12, 223-246, 1974.&lt;/p&gt;<a name="line.363"></a>
<FONT color="green">364</FONT>         *<a name="line.364"></a>
<FONT color="green">365</FONT>         * &lt;p&gt;For shape &gt;= 1: &lt;br/&gt;<a name="line.365"></a>
<FONT color="green">366</FONT>         * Marsaglia and Tsang, &lt;i&gt;A Simple Method for Generating<a name="line.366"></a>
<FONT color="green">367</FONT>         * Gamma Variables.&lt;/i&gt; ACM Transactions on Mathematical Software,<a name="line.367"></a>
<FONT color="green">368</FONT>         * Volume 26 Issue 3, September, 2000.&lt;/p&gt;<a name="line.368"></a>
<FONT color="green">369</FONT>         *<a name="line.369"></a>
<FONT color="green">370</FONT>         * @return random value sampled from the Gamma(shape, scale) distribution<a name="line.370"></a>
<FONT color="green">371</FONT>         */<a name="line.371"></a>
<FONT color="green">372</FONT>        @Override<a name="line.372"></a>
<FONT color="green">373</FONT>        public double sample()  {<a name="line.373"></a>
<FONT color="green">374</FONT>            if (shape &lt; 1) {<a name="line.374"></a>
<FONT color="green">375</FONT>                // [1]: p. 228, Algorithm GS<a name="line.375"></a>
<FONT color="green">376</FONT>    <a name="line.376"></a>
<FONT color="green">377</FONT>                while (true) {<a name="line.377"></a>
<FONT color="green">378</FONT>                    // Step 1:<a name="line.378"></a>
<FONT color="green">379</FONT>                    final double u = random.nextDouble();<a name="line.379"></a>
<FONT color="green">380</FONT>                    final double bGS = 1 + shape / FastMath.E;<a name="line.380"></a>
<FONT color="green">381</FONT>                    final double p = bGS * u;<a name="line.381"></a>
<FONT color="green">382</FONT>    <a name="line.382"></a>
<FONT color="green">383</FONT>                    if (p &lt;= 1) {<a name="line.383"></a>
<FONT color="green">384</FONT>                        // Step 2:<a name="line.384"></a>
<FONT color="green">385</FONT>    <a name="line.385"></a>
<FONT color="green">386</FONT>                        final double x = FastMath.pow(p, 1 / shape);<a name="line.386"></a>
<FONT color="green">387</FONT>                        final double u2 = random.nextDouble();<a name="line.387"></a>
<FONT color="green">388</FONT>    <a name="line.388"></a>
<FONT color="green">389</FONT>                        if (u2 &gt; FastMath.exp(-x)) {<a name="line.389"></a>
<FONT color="green">390</FONT>                            // Reject<a name="line.390"></a>
<FONT color="green">391</FONT>                            continue;<a name="line.391"></a>
<FONT color="green">392</FONT>                        } else {<a name="line.392"></a>
<FONT color="green">393</FONT>                            return scale * x;<a name="line.393"></a>
<FONT color="green">394</FONT>                        }<a name="line.394"></a>
<FONT color="green">395</FONT>                    } else {<a name="line.395"></a>
<FONT color="green">396</FONT>                        // Step 3:<a name="line.396"></a>
<FONT color="green">397</FONT>    <a name="line.397"></a>
<FONT color="green">398</FONT>                        final double x = -1 * FastMath.log((bGS - p) / shape);<a name="line.398"></a>
<FONT color="green">399</FONT>                        final double u2 = random.nextDouble();<a name="line.399"></a>
<FONT color="green">400</FONT>    <a name="line.400"></a>
<FONT color="green">401</FONT>                        if (u2 &gt; FastMath.pow(x, shape - 1)) {<a name="line.401"></a>
<FONT color="green">402</FONT>                            // Reject<a name="line.402"></a>
<FONT color="green">403</FONT>                            continue;<a name="line.403"></a>
<FONT color="green">404</FONT>                        } else {<a name="line.404"></a>
<FONT color="green">405</FONT>                            return scale * x;<a name="line.405"></a>
<FONT color="green">406</FONT>                        }<a name="line.406"></a>
<FONT color="green">407</FONT>                    }<a name="line.407"></a>
<FONT color="green">408</FONT>                }<a name="line.408"></a>
<FONT color="green">409</FONT>            }<a name="line.409"></a>
<FONT color="green">410</FONT>    <a name="line.410"></a>
<FONT color="green">411</FONT>            // Now shape &gt;= 1<a name="line.411"></a>
<FONT color="green">412</FONT>    <a name="line.412"></a>
<FONT color="green">413</FONT>            final double d = shape - 0.333333333333333333;<a name="line.413"></a>
<FONT color="green">414</FONT>            final double c = 1 / (3 * FastMath.sqrt(d));<a name="line.414"></a>
<FONT color="green">415</FONT>    <a name="line.415"></a>
<FONT color="green">416</FONT>            while (true) {<a name="line.416"></a>
<FONT color="green">417</FONT>                final double x = random.nextGaussian();<a name="line.417"></a>
<FONT color="green">418</FONT>                final double v = (1 + c * x) * (1 + c * x) * (1 + c * x);<a name="line.418"></a>
<FONT color="green">419</FONT>    <a name="line.419"></a>
<FONT color="green">420</FONT>                if (v &lt;= 0) {<a name="line.420"></a>
<FONT color="green">421</FONT>                    continue;<a name="line.421"></a>
<FONT color="green">422</FONT>                }<a name="line.422"></a>
<FONT color="green">423</FONT>    <a name="line.423"></a>
<FONT color="green">424</FONT>                final double x2 = x * x;<a name="line.424"></a>
<FONT color="green">425</FONT>                final double u = random.nextDouble();<a name="line.425"></a>
<FONT color="green">426</FONT>    <a name="line.426"></a>
<FONT color="green">427</FONT>                // Squeeze<a name="line.427"></a>
<FONT color="green">428</FONT>                if (u &lt; 1 - 0.0331 * x2 * x2) {<a name="line.428"></a>
<FONT color="green">429</FONT>                    return scale * d * v;<a name="line.429"></a>
<FONT color="green">430</FONT>                }<a name="line.430"></a>
<FONT color="green">431</FONT>    <a name="line.431"></a>
<FONT color="green">432</FONT>                if (FastMath.log(u) &lt; 0.5 * x2 + d * (1 - v + FastMath.log(v))) {<a name="line.432"></a>
<FONT color="green">433</FONT>                    return scale * d * v;<a name="line.433"></a>
<FONT color="green">434</FONT>                }<a name="line.434"></a>
<FONT color="green">435</FONT>            }<a name="line.435"></a>
<FONT color="green">436</FONT>        }<a name="line.436"></a>
<FONT color="green">437</FONT>    }<a name="line.437"></a>




























































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